Abstract

We propose a combination of ray optics and Fraunhofer multiple-slit diffraction theory for calculating the two-dimensional triangular periodic grating in the resonance domain. The peak of the envelope pattern of angular distribution of diffraction efficiency is calculated by ray optics while the peak width is calculated using Fraunhofer theory. It was clarified, using rigorous coupled-wave analysis and a nonstandard-finite-difference time-domain method, that the envelope pattern of the diffraction of the grating could be calculated easily and understood intuitively for the design of displays and lighting.

Figures (11)

Shape of the triangular grating profile and the definition of Λ, d, and ds. The fill factor is 0.5. The light direction for case A is shown; ds is assumed to be infinity. The refractive index n of the grating is 1.5.

Angular distribution of the diffraction efficiency for different polarization and light directions; Λ∕λ is 9.1 or 22.7. The aspect ratio is 1. Though the diffraction efficiency is discrete, it is connected with the auxiliary line to make it intelligible.

Angular distribution of the transmissive diffraction efficiency of the TE mode for case B for different Λ∕λ. The aspect ratio is 1. The diffraction efficiency is connected with the auxiliary line to make it intelligible; Λ∕λ is varied from 4.5 to 22.7.

Angular distribution of the transmissive diffraction efficiency of the TE mode for case A for different Λ∕λ. The aspect ratio is 2. The diffraction efficiency is connected with the auxiliary line to make it intelligible; Λ∕λ is varied from 4.5 to 22.7.

Angular distribution of the transmissive diffraction efficiency of the TE mode for different d∕Λ for case A. The diffraction efficiency is connected with the auxiliary line to make it intelligible. (a) d∕Λ is varied from 0.25 to 3. (b) d∕Λ is 2.5 and 3 and Λ∕λ is 9.1 and 22.7.

Angular distribution of the transmitted light is shown. The incidence angle is 20°, the direction of the incident light is case B, the polarization is TE, d∕Λ is 1, and Λ∕λ was varied. The angle of the peak is near 90°. The angular distribution varies greatly with Λ∕λ.

Field for FDTD calculation and the grating. The black rectangular area is expanded into the area of (b). (b) Phase distribution of the scattered light in the TE mode and for case A. There is only one groove, unlike Fig. 1. The width of the groove is 9.1λ and the aspect ratio is 1.

Geometry of forward-diffracted wave vectors showing the conical nature of diffraction. Forward-diffracted waves (i=−1 to i=+2) are indicated by the arrow. The light travels from region 1(z<0) to region 3(z>d).